Question

A cylinder is filled with a liquid of density $ P = \dfrac{F}{A} $ up to a height $ 'h' $ . If the beaker is at rest, then determine the mean pressure on the wall is?

Answer 1

Hint: In this question, first we need to calculate the pressure at the top of the beaker and then we will calculate the pressure at the bottom of the beaker. With the help of the pressure at the top and at the bottom, we will calculate the mean pressure on the wall.

Complete answer:
Whenever a fluid exerts pressure on an object, the magnitude of the pressure is the lowest at the top of the object and the greatest at the bottom. To understand this concept, let us take the example of a transparent container filled with water.
By only looking at a transparent container filled with water, we can see that the transparent container will have more water molecules at the bottom than at the top. By sheer weight, pressure, which is defined as the force exerted on a given area, is more at the bottom because the force acting here is more.
We know that the formula for pressure is,
 $ P = \rho gh $
Where,
 $ P $ is the pressure exerted
 $ \rho $ is the density of the fluid
 $ g $ is the acceleration due to gravity
 $ h $ is the depth of the fluid
Pressure at the top of the cylinder,
  $ {P_1} = d \times g \times 0 $ (as the depth at the top is $ 0 $ )
 $ {P_1} = 0 $
Pressure at the bottom of the cylinder,
 $ {P_2} = d \times g \times h $ (as the depth at the bottom is $ h $ )
 $ {P_2} = dgh $
To calculate the mean pressure, we will take the average of $ {P_1} $ and $ {P_2} $
Let the mean pressure be $ P $
 $ P = \dfrac{{{P_1} + {P_2}}}{2} $
 $ P = \dfrac{{0 + dgh}}{2} $
On simplifying, we get,
 $ P = \dfrac{{dgh}}{2} $
So, the final answer is $ P = \dfrac{{dgh}}{2} $ .

Note:
It is important to note that the formula $ P = \rho gh $ is a special case of the formula $ P = \dfrac{F}{A} $ . The general formula for pressure is $ P = \dfrac{F}{A} $ . The formula $ P = \rho gh $ is used to calculate the hydrostatic pressure. Hydrostatic pressure refers to the pressure that any fluid in a confined space exerts.